The isochron method

Many radioactive dating methods are based on minute additions of daughter products to a rock or mineral in which a considerable amount of daughter-type isotopes already exists. These isotopes did not come from radioactive decay in the system but rather formed during the original creation of the elements. In this case, it is a big advantage to present the data in a form in which the abundance of both the parent and daughter isotopes are given with respect to the abundance of the initial background daughter. The incremental additions of the daughter type can then be viewed in proportion to the abundance of parent atoms. In mathematical terms this is achieved as follows. It has already been shown—7—that the number of daughter atoms present from radioactive decay D* can be related to the number of parent atoms remaining P by the simple expression:

When some daughter atoms are initially present (designated D0), the total number D is the sum of radiogenic and initial atoms, so that

To establish the condition that both parent and daughter abundances should be relative to the initial background, a stable isotope S of the daughter element can be chosen and divided into all portions of this equation; thus,

This equation has the form; y = b + xm, which is that of a straight line on x–y coordinates. The slope m is equal to (eλt − 1) and the intercept is equal to (D/S)0. This term, shown in Figure 1, is called the initial ratio. The slope is proportional to the geologic age of the system.

In practice, the isochron approach has many inherent advantages. When a single body of liquid rock crystallizes, parent and daughter elements may separate so that, once solid, the isotopic data would define a series of points, such as those shown as open circles designated R1, R2, R3 in Figure 1. They plot along a horizontal line reflecting a common value for the initial daughter isotope ratio (D/S)0. With time each would then develop additional daughter abundances in proportion to the amount of parent present. If a number of samples are analyzed and the results are shown to define a straight line within error, then a precise age is defined because this is only possible if each is a closed system and each has the same initial ratio and age. The uncertainty in determining the slope is reduced because it is defined by many points. A second advantage of the method relates to the fact that under high-temperature conditions the daughter isotopes may escape from the host minerals. In this case, a valid age can still be obtained, provided that they remain within the rock. Should a point plot below the line, it could indicate that a particular sample was open to migration of the dating elements or that the sample was contaminated and lay below the isochron when the rock solidified.

Rubidium–strontium (Rb–Sr) dating was the first technique in which the whole rock isochron method was extensively employed. Certain rocks that cooled quickly at the surface were found to give precisely defined linear isochrons, but many others did not. Some studies have shown that rubidium is very mobile both in fluids that migrate through the rock as it cools and in fluids that are present as the rock undergoes chemical weathering. Similar studies have shown that the samarium–neodymium (Sm–Nd) parent–daughter pair is more resistant to secondary migration but that, in this instance, sufficient initial spread in the abundance of the parent isotope is difficult to achieve.

Analysis of separated minerals

When an igneous rock crystallizes, a wide variety of major and trace minerals may form, each concentrating certain elements and radioactive trace elements within the rock. By careful selection, certain minerals that contain little or no daughter element but abundant parent element can be analyzed. In this case, the slope of the line in Figure 1 is computed from an assumed value for the initial ratio, and it is usually possible to show that uncertainties related to this assumption are negligible. This is possible in potassium–argon (K–Ar) dating, for example, because most minerals do not take argon into their structures initially. In rubidium–strontium dating, micas exclude strontium when they form, but accept much rubidium. In uranium–lead (U–Pb) dating of zircon, the zircon is found to exclude initial lead almost completely. Minerals, too, are predictable chemical compounds that can be shown to form at specific temperatures and remain closed up to certain temperatures if a rock has been reheated or altered. A rock, on the other hand, may contain minerals formed at more than one time under a variety of conditions. Under such circumstances the isolation and analysis of certain minerals can indicate at what time these conditions prevailed. If a simple mineral is widespread in the geologic record, it is more valuable for dating as more units can be measured for age and compared by the same method. However, if a single parent–daughter pair that is amenable to precise analysis can be measured in a variety of minerals, the ages of a wide variety of rock types can be determined by a single method without the need for intercalibration. In some cases the discovery of a rare trace mineral results in a major breakthrough as it allows precise ages to be determined in formerly undatable units. For example, the mineral baddeleyite, an oxide of zirconium (ZrO2), has been shown to be widespread in small amounts in mafic igneous rocks (i.e., those composed primarily of one or more ferromagnesian, dark-coloured minerals). Here, a single uranium–lead isotopic analysis can provide an age more precise than can be obtained by the whole rock isochron method involving many analyses. When single minerals are analyzed, each grain can be studied under a microscope under intense side light so that alterations or imperfections can be revealed and excluded. If minerals are used for dating, the necessary checks on the ages are achieved by analyzing samples from more than one location and by analyzing different grain sizes or mineral types that respond differently to disturbing events. It can be said that minerals provide a high degree of sample integrity that can be predicted on the basis of experience gained through numerous investigations under a variety of geologic conditions. An ideal mineral is one that has sufficient parent and daughter isotopes to measure precisely, is chemically inert, contains little or no significant initial daughter isotopes, and retains daughter products at the highest possible temperatures. A specific datable mineral like rutile, which can be linked to a specific event such as the formation of a mineral deposit, is especially important.

Since the Earth was formed, the abundance of daughter product isotopes has increased through time. For example, the ratio of lead of mass 206 relative to that of mass 204 has changed from an initial value of about 10 present when the Earth was formed to an average value of about 19 in rocks at the terrestrial surface today. This is true because uranium is continuously creating more lead. A lead-rich mineral formed and isolated early in Earth history would have a low lead-206 to lead-204 ratio because it did not receive subsequent additions by the radioactive decay of uranium. If the Earth’s interior were a simple and homogeneous reservoir with respect to the ratio of uranium to lead, a single sample extracted by a volcano would provide the time of extraction. This would be called a model age. No parent–daughter value for a closed system is involved, rather just a single isotopic measurement of lead viewed with respect to the expected evolution of lead in the Earth. Unfortunately the simplifying assumption in this case is not true, and lead model ages are approximate at best. Other model ages can be calculated using neodymium isotopes by extrapolating present values back to a proposed mantle-evolution line. In both cases, approximate ages that have a degree of validity with respect to one another result, but they are progressively less reliable as the assumptions on which the model is calculated are violated.

The progressive increase in the abundance of daughter isotopes over time gains a special significance where the parent element is preferentially enriched in either the mantle or the crust. For example, rubidium is concentrated in the crust, and as a result the present-day continents, subjected to weathering, have an elevated radiogenic to stable isotope ratio (87Sr/86Sr) of 0.720. In contrast, modern volcanic rocks in the oceans imply that much of the mantle has a value between about 0.703 and 0.705. Should crustal material be recycled, the strontium isotopic signature of the melt would be diagnostic.

Fossils record the initial, or primary, age of a rock unit. Isotopic systems, on the other hand, can yield either the primary age or the time of a later event, because crystalline materials are very specific in the types of atoms they incorporate, in terms of both the atomic size and charge. An element formed by radioactive decay is quite different from its parent atom and thus is out of place with respect to the host mineral. All it takes for such an element to be purged from the mineral is sufficient heat to allow solid diffusion to occur. Each mineral has a temperature at which rapid diffusion sets in, so that, as a region is slowly heated, first one mineral and then another loses its daughter isotopes. When this happens, the isotopic “clock” is reset to zero, where it remains until the mineral cools below the blocking temperature. (This is the temperature below which a mineral becomes a closed chemical system for a specific radioactive decay series. Accordingly, the parent–daughter isotope ratio indicates the time elapsed since that critical threshold was reached.) In this case, the host mineral could have an absolute age very much older than is recorded in the isotopic record. The isotopic age then is called a cooling age. It is even possible by using a series of minerals with different blocking temperatures to establish a cooling history of a rock body—i.e., the times since the rock body cooled below successively lower temperatures. Such attempts can be complicated by the fact that a mineral may “grow” below the blocking temperature rather than simply become closed to isotopic migration. When this happens, the age has little to do with the cooling time. Another problem arises if a region undergoes a second reheating event. Certain minerals may record the first event, whereas others may record the second, and any suggestion of progressive cooling between the two is invalid. This complication does not arise when rapid cooling has occurred. Identical ages for a variety of minerals with widely different blocking temperatures is unequivocal proof of rapid cooling.

Fortunately for geologists the rock itself records in its texture and mineral content the conditions of its formation. A rock formed at the surface with no indication of deep burial or new mineral growth can be expected to give a valid primary age by virtue of minerals with low blocking temperatures. On the other hand, low-blocking-point minerals from a rock containing minerals indicative of high temperatures and pressures cannot give a valid primary age. Such minerals would be expected to remain open until deep-level rocks of this sort were uplifted and cooled.

Given these complicating factors, one can readily understand why geochronologists spend a great deal of their time and effort trying to see through thermal events that occurred after a rock formed. The importance of identifying and analyzing minerals with high blocking temperatures also cannot be overstated. Minerals with high blocking temperatures that form only at high temperatures are especially valuable. Once formed, these minerals can resist daughter loss and record the primary age even though they remained hot (say, 700° C) for a long time. The mineral zircon datable by the uranium–lead method is one such mineral. The mica mineral biotite dated by either the potassium–argon or the rubidium–strontium method occupies the opposite end of the spectrum and does not retain daughter products until cooled below about 300° C. Successively higher blocking temperatures are recorded for another mica type known as muscovite and for amphibole, but the ages of both of these minerals can be completely reset at temperatures that have little or no effect on zircon.

Taken in perspective, it is evident that many parts of the Earth’s crust have experienced reheating temperatures above 300° C—i.e., reset mica ages are very common in rocks formed at deep crustal levels. Vast areas within the Precambrian shield, which have identical ages reflecting a common cooling history, have been identified. These are called geologic provinces. By contrast, rocks that have approached their melting point, say, 750° C, which can cause new zircon growth during a second thermal event, are rare, and those that have done this more than once are almost nonexistent.

Instruments and procedures

Use of mass spectrometers

The age of a geologic sample is measured on as little as a billionth of a gram of daughter isotopes. Moreover, all the isotopes of a given chemical element are nearly identical except for a very small difference in mass. Such conditions necessitate instrumentation of high precision and sensitivity. Both these requirements are met by the modern mass spectrometer. A high-resolution mass spectrometer of the type used today was first described by the American physicist Alfred O. Nier in 1940, but it was not until about 1950 that such instruments became available for geochronological research.

For isotopic dating with a mass spectrometer, a beam of charged atoms, or ions, of a single element from the sample is produced. This beam is passed through a strong magnetic field in a vacuum, where it is separated into a number of beams, each containing atoms of only the same mass. Because of the unit electric charge on every atom, the number of atoms in each beam can be evaluated by collecting individual beams sequentially in a device called a Faraday cup. Once in this collector, the current carried by the atoms is measured as it leaks across a resistor to ground. Currents measured are small, only from 10−11 to 10−15 ampere, so that shielding and preamplification are required as close to the Faraday cup as possible. It is not possible simply to count the atoms, because all atoms loaded into the source do not form ions and some ions are lost in transmission down the flight tube. Precise and accurate information as to the number of atoms in the sample can, however, be obtained by measuring the ratio of the number of atoms in the various separated beams. By adding a special artificially enriched isotope during sample dissolution and by measuring the ratio of natural to enriched isotopes in adjacent beams, the number of daughter isotopes can be readily determined. The artificially enriched isotope is called a “spike.” It is usually a highly purified form of a low-abundance natural isotope, but an even better spike is an isotope with a mass not found in nature at all. Lead-205 produced in a type of particle accelerator called a cyclotron constitutes such an ideal spike.

As the sample is heated and vaporizes under the vacuum in the source area of the mass spectrometer, it is commonly observed that the lighter isotopes come off first, causing a bias in the measured values that changes during the analysis. In most cases this bias, or fractionation, can be corrected if the precise ratio of two of the stable isotopes present is known. Today’s state-of-the-art instruments produce values for strontium and neodymium isotopic abundances that are reproducible at a level of about 1 in 20,000. Such precision is often essential in the isochron method (see above) because of the small changes in relative daughter abundance that occur over geologic time.

Technical advances

The ability to add a single artificial mass to the spectrum in a known amount and to determine the abundances of other isotopes with respect to this provides a powerful analytical tool. By means of this process known as isotope dilution, invisibly small amounts of material can be analyzed, and because only ratios are involved, a loss of part of the sample during preparation has no effect on the result. Spike solutions can be calibrated simply by obtaining a highly purified form of the element being calibrated. After carefully removing surface contamination, a precisely weighted portion of the element is dissolved in highly purified acid and diluted to the desired level in a weighed quantity of water. What is required is dilution of one cubic centimetre to a litre from which a second cubic centimetre is again diluted to a litre to approach the range of parts per million or parts per billion typically encountered in samples. In this way, a known number of natural isotopes can be mixed with a known amount of spike and the concentration in the spike solution determined from the ratio of the masses. Once the calibration has been completed, the process is reversed and a weighed amount of spike is mixed with the parent and daughter elements from a mineral or rock. The ratio of the masses then gives the number of naturally produced atoms in the sample. The use of calibrated enriched isotopic tracers facilitates checks for contamination, even though the process is time-consuming. A small but known amount of tracer added to a beaker of water can be evaporated under clean-room conditions. Once loaded in a mass spectrometer, the contamination from the beaker and the water is easily assessed with respect to the amount of spike added. Contamination as small as 10−12 gram can be detected by this method.

The materials analyzed during isotopic investigations vary from microgram quantities of highly purified mineral grains to gram-sized quantities of rock powders. In all cases, the material must be dissolved without significant contamination. The spike should be added before dissolution. Most of the minerals in rocks can be dissolved in a day or so at a temperature near 100° C. Certain minerals that are highly refractory both in nature and in the laboratory (e.g., zircon) may require five days or more at temperatures near 220° C. In this case, the sample is confined in a solid Teflon (trade name for a synthetic resin composed of polytetrafluoroethylene), metal-clad pressure vessel, introduced by the Canadian geochronologist Thomas E. Krogh in 1973.

The method just described proved to be a major technical breakthrough as it resulted in a reduction in lead-background contamination by a factor of between 10,000 and nearly 1,000,000. This means that a single grain can now be analyzed with a lower contamination level (or background correction) than was possible before with 100,000 similar grains. Advances in high-sensitivity mass spectrometry of course were essential to this development.

Once dissolved, the sample is ready for the chemical separation of the dating elements. This is generally achieved by using the methods of ion-exchange chromatography. In this process, ions are variously adsorbed from solution onto materials with ionic charges on their surface and separated from the rest of the sample. After the dating elements have been isolated, they are loaded into a mass spectrometer and their relative isotopic abundances determined.

The abundance of certain isotopes used for dating is determined by counting the number of disintegrations per minute (i.e., emission activity). The rate is related to the number of such atoms present through the half-life. For example, a certain amount of carbon-14 (14C) is present in all biological components at the Earth’s surface. This radioactive carbon is continually formed when nitrogen atoms of the upper atmosphere collide with neutrons produced by the interaction of high-energy cosmic rays with the atmosphere. An organism takes in small amounts of carbon-14, together with the stable (nonradioactive) isotopes carbon-12 (12C) and carbon-13 (13C), as long as it is alive. Once it dies, however, no additional carbon-14 is acquired and the level of radiocarbon in the organism’s tissue decreases progressively as a function of half-life. The time that has passed since the organism was alive can be determined by counting the beta emissions from a tissue sample. The number of emissions in a given time period is proportional to the amount of residual carbon-14.

The introduction of an instrument called an accelerator mass spectrometer has brought about a major advance in radiocarbon dating. Unlike the old detector (e.g., the Geiger counter) that counts the few decay particles emitted from a large amount of carbon, the new instrument counts directly all of the carbon-14 atoms in a sample. This increase in instrument sensitivity has made it possible to reduce the sample size by as much as 10,000 times and at the same time improve the precision of ages measured. (For a detailed discussion of radiocarbon age determination, see below Carbon-14 dating and other cosmogenic methods.)

In a similar development, the use of highly sensitive thermal ionization mass spectrometers is replacing the counting techniques employed in some disequilibrium dating (see below). Not only has this led to a reduction in sample size and measurement errors but it also has permitted a whole new range of problems to be investigated. Certain parent–daughter isotopes are extremely refractory and do not ionize in a conventional mass spectrometer. To solve this problem, researchers are developing new instruments in which a small amount of material can be evaporated from the surface with a pulse of energy and ionized with a pulse of laser light. A major trend anticipated in geochronology and isotope geochemistry involves the analysis of mineral grains in place without chemical dissolution and mass spectrometry. This type of analysis requires expensive equipment in which a focused beam of ions is directed at a spot on a mineral sample. This causes atoms to evaporate from the surface, and the ions produced are extracted and measured in a mass spectrometer. Uranium–lead dating of zircon by this method has been pioneered by William Compston at the Australian National University.

Major methods of isotopic dating

Isotopic dating relative to fossil dating requires a great deal of effort and depends on the integrated specialized skills of geologists, chemists, and physicists. It is, nevertheless, a valuable resource that allows correlations to be made over virtually all of Earth history with a precision once only possible with fossiliferous units that are restricted to the most recent 12 percent or so of geologic time. Although any method may be attempted on any unit, the best use of this resource requires that every effort be made to tackle each problem with the most efficient technique. Because of the long half-life of some isotopic systems or the high background or restricted range of parent abundances, some methods are inherently more precise. The skill of a geochronologist is demonstrated by the ability to attain the knowledge required and the precision necessary with the least number of analyses. The factors considered in selecting a particular approach are explored here.

Uranium–lead method

As each dating method was developed, tested, and improved, mainly since 1950, a vast body of knowledge about the behaviour of different isotopic systems under different geologic conditions has evolved. It is now clear that with recent advances the uranium–lead method is superior in providing precise age information with the least number of assumptions. The method has evolved mainly around the mineral zircon (ZrSiO4). Because of the limited occurrence of this mineral, it was once true that only certain felsic igneous rocks (those consisting largely of the light-coloured, silicon and aluminum-rich minerals feldspar and quartz) could be dated. Today, however, baddeleyite (ZrO2) and zirconolite (CaZrTi2O7) have been found to be widespread in the silica-poor mafic igneous rocks. In addition, perovskite (CaTiO3), a common constituent of some ultramafic igneous rocks, has been shown to be amenable to precise uranium–lead dating. As a result of these developments, virtually all igneous rocks can now be dated. This capability, moreover, has been enhanced because the most advanced geochronological laboratories are able to analyze samples that weigh only a few millionths of a gram. This amount can be found in a comparatively large number of rocks, whereas the amount previously required (about 0.1 gram) cannot. Age determinations also can now be made of low-uranium trace minerals such as rutile (TiO2), a common constituent found in mineral deposits, adding still further to the number of entities that are datable by the uranium–lead method. Other minerals commonly employed to date igneous and metamorphic rocks include titanite, monazite, and even garnet in certain favourable cases. Additional minerals have been tried with varying success.

Double uranium-lead chronometers

The reason why uranium–lead dating is superior to other methods is simple: there are two uranium–lead chronometers. Because there exist two radioactive uranium atoms (those of mass 235 and 238), two uranium–lead ages can be calculated for every analysis. The age results or equivalent daughter–parent ratio can then be plotted one against the other on a concordia diagram. If the point falls on the upper curve shown, the locus of identical ages, the result is said to be concordant, and a closed-system unequivocal age has been established. Any leakage of daughter isotopes from the system will cause the two ages calculated to differ, and data will plot below the curve. Because each of the daughters has a different half-life, early leakage will affect one system more than the other. Thus, there is a built-in mechanism that can prove or disprove whether a valid age has been measured. Historically, it had been observed that the uranium–lead systems in the mineral zircon from unmetamorphosed rocks were almost invariably disturbed or discordant but yielded a linear array on the concordia diagram. Given a set of variably disturbed samples, an extrapolation to zero disturbance was possible. More recently, it has been found that of all the grains present in a rock a very few still retain closed isotopic systems but only in their interior parts. Thus, grains with a diameter comparable to that of a human hair, selected under a microscope to be crack-free and of the highest possible quality, have been found to be more concordant than cracked grains. In addition, it has been shown that most such grains can be made much more concordant by mechanically removing their outer parts using an air-abrasion technique. Of course, the ability to analyze samples weighing only a few millionths of a gram was essential to this development. As noted earlier, this in turn was possible solely because the lead background contamination had been reduced from 1 × 10−6 gram to almost 1 × 10−12 gram per analysis. The methods of selection and abrasion used to locate grains with closed isotopic systems could be worked out only because the uranium–lead method has the inherent ability to assess with a single analysis whether or not a closed isotopic system has prevailed.

The presence of two radioactive parents provides a second major advantage because, as daughter products, lead atoms are formed at different rates and their relative abundance undergoes large changes as a function of time. Thus,the ratio of lead-207 to lead-206 changes by about 0.1 percent every two million years. Since this ratio is easily calibrated and reproduced at such a level of precision, errors as low as ±2 million years at a confidence level of 95 percent are routinely obtained on lead-207–lead-206 ages. By contrast, errors as high as ±30 to 50 million years are usually quoted for the rubidium–strontium and samarium–neodymium isocron methods (see belowRubidium–strontium method; Samarium–neodymium method).